Linear Parameter-Varying Control of Nonlinear Systems PDF

i i i i I Automotive um ul...

i i i i I Automotive um ul rr ic u 001 C Y N S U C I-1 i i  i i i i 1 Linear Parameter-Varying Control of Nonlinear Systems with Applications to Automotive um and ul Aerospace Controls ∗ rr ic u 002 C 1.1 Introduction........................................................ 1-1 1.2 Statement of the Control Problem..................... 1-2 Y 1.3 LPV H∞ Control................................................ 1-3 N 1.4 Choosing the LPV Weights W (θ, s)................. 1-6 1.5 Handling PV Time Delays................................. 1-7 S U 1.6 Applications in Automotive Engine Control.... 1-8 Feedback Fuel Control Feedforward Fuel Control C 1.7 Application in Aircraft Flight Control............ 1-10 Hans P. Geering 1.8 Conclusions....................................................... 1-11 Swiss Federal Institute of Technology References.................................................................... 1-12 1.1 Introduction In this chapter, a linear parameter-varying (LPV) plant [A(θ), B(θ), C(θ)] with the parameter vector θ is considered with continuously differentiable system matrices A(θ), B(θ), and C(θ). As described in Section 1.2, such a LPV plant description is typically obtained by linearizing the model of a nonlinear plant about a nominal trajectory. The control problem, which is considered in this chapter is finding a LPV continuous-time controller with the system matrices F(θ), G(θ), and H(θ) of its state–space model. In Section 1.3, the control problem is formulated as an H∞ problem using the mixed sensitivity approach. The shaping weights We (θ, s), Wu (θ, s), and Wy (θ, s) are allowed to be parameter-varying. The most appealing feature of this approach is that it yields a parameter-varying bandwidth ωc (θ) of the robust control system. Choosing appropriate shaping weights is described in Section 1.4. For more details about the design methodology, the reader is referred to [1–6]. ∗ Parts reprinted from H. P. Geering, Proceedings of the IEEE International Symposium on Industrial Electronics—ISIE 2005, Dubrovnik, Croatia, June 20–23, 2005, pp. 241–246, © 2005. IEEE. With permission. 1-1 i i  i i i i 1-2 Control System Applications In Section 1.5, it is shown, how parameter-varying time-delays in the plant dynamics can be handled in the framework proposed in Sections 1.3 and 1.4. For more details, consult [5,7]. In Section 1.6, two applications in the area of automotive engine control are discussed. In the first application [4,5,8], the design of an LPV feedback controller for the fuel injection is shown, which is suitable over the whole operating envelope of engine. In the second application [9,10], the philosophy of designing an LPV feedback controller is carried over to the problem of designing an additional LPV feedforward controller compensating the parameter- varying wall-wetting dynamics in the intake manifold of the port-injected gasoline engine. In Section 1.7, the problem of LPV control of the short-period motion of an aircraft is discussed. 1.2 Statement of the Control Problem We consider the following nonlinear time-invariant dynamic system (“plant”) with the unconstrained input vector U(t) ∈ Rm , the state vector X(t) ∈ Rn , and the output vector Y (t) ∈ Rp : Ẋ(t) = f (X(t), U(t)), Y (t) = g(X(t)), um where f and g are fairly “smooth” continuously differentiable functions. Let us assume that we have found a reasonable or even optimal open-loop control strategy Unom (t) for a rather large time interval t ∈ [0, T] (perhaps T = ∞), which theoretically generates the nominal state and output trajectories Xnom (t) and Ynom (t), respectively. ul ric In order to ensure that the actual state and output trajectories X(t) and Y (t) stay close to the nominal ones at all times, we augment the open-loop control Unom (t) with a (correcting) feedback part u(t). Thus, ur the combined open-closed-loop input vector becomes 003 C U(t) = Unom (t) + u(t). Y Assuming that the errors N x(t) = X(t) − Xnom (t) and y(t) = Y (t) − Ynom (t) SU of the state and output trajectories, respectively, can be kept minimum with small closed-loop corrections u(t), allows us to design a linear (parameter-varying) output feedback controller based on the linearized C dynamics of the plant: ẋ(t) = A(θ)x(t) + B(θ)u(t), y(t) = C(θ)x(t), where A(θ), B(θ), and C(θ) symbolically denote the following Jacobi matrices: ∂f   A(θ) = Xnom (t), Unom (t) , ∂x ∂f   B(θ) = Xnom (t), Unom (t) , ∂u ∂g   C(θ) = Xnom (t). ∂x The symbol θ (or more precisely θ(t)) denotes a parameter vector, by which the Jacobi matrices are parametrized; it contains the reference values Xnom (t) and Unom (t) of the state and control vector, respectively, but it may also contain additional “exogenous” signals influencing the parameters of the i i  i i i i Linear Parameter-Varying Control of Nonlinear Systems 1-3 r e us ys K Gs FIGURE 1.1 Schematic representation of the feedback control system. nonlinear equations describing the dynamics of the plant (e.g., a temperature, which is not included in the model as a state variable). By using the symbol θ rather than θ(t), we indicate that we base the design of the feedback controller on a time-invariant linearized plant at every instant t (“frozen linearized dynamics”). This leads us to posing the following problem of designing an LPV controller: For all of the attainable values of the parameter vector θ, design a robust dynamic controller (with a suitable order nc ) with the state–space representation z(t) ∈ Rnc , ż(t) = Ac (θ)z(t) + Bc (θ)e(t), us (t) = Cc (θ)z(t), such that all of the specified quantitative, parameter-dependent performance, and robustness specifica- um tions are met (see Figure 1.1). In Section 1.3, this rather general problem statement will be narrowed down to a suitable and trans- parent setting of H∞ control and the solution will be presented. ul ric 1.3 LPV H∞ Control ur 004 In this section, we consider the LPV time-invariant plant C ẋs (t) = As (θ)xs (t) + Bs (θ)us (t), Y ys (t) = Cs (θ)xs (t) N SU of order ns. For the sake of simplicity, we assume that we have a “square” plant, that is, the number of output signals equals the number of input signals: ps = ms. Furthermore, we assume that the input us , the state xs , and the output ys are suitably scaled, such that C the singular values of the frequency response matrix Gs (jω) = Cs [jωI − As ]−1 Bs are not spread too wide apart. For the design of the LPV time-invariant controller K(θ) depicted in Figure 1.1, we use the H∞ method [1,2]. As a novel feature, we use parameter-dependent weights W (θ, s). This allows in particular that we can adapt the bandwidth ωc (θ) of the closed-loop control system to the parameter-dependent properties of the plant! Figure 1.2 shows the abstract schematic of the generic H∞ control system. Again, K(θ) is the controller, which we want to design and G(θ, s) is the so-called augmented plant. The goal of the design is finding a compensator K(θ, s), such that the H∞ norm from the auxiliary input w to the auxiliary output z is less w z G us e K FIGURE 1.2 Schematic representation of the H∞ control system. i i  i i i i 1-4 Control System Applications We ze Wu zu w K Gs Wy zy − FIGURE 1.3 S/KS/T weighting scheme. than γ (γ ≤ 1), that is, Tzw (θ, s)∞ < γ ≤ 1 for all of the attainable values of the constant parameter vector θ. For the H∞ design we choose the mixed-sensitivity approach. This allows us to shape the singular values of the sensitivity matrix S( jω) and of the complementary sensitivity matrix T( jω) of our control system (Figure 1.1), where S(θ, s) = [I + Gs (θ, s)K(θ, s)]−1 T(θ, s) = Gs (θ, s)K(θ, s)[I + Gs (θ, s)K(θ, s)]−1 um = Gs (θ, s)K(θ, s)S(θ, s). ul Thus, we choose the standard S/KS/T weighting scheme as depicted in Figure 1.3. This yields the following transfer matrix: ric ⎡ ⎤ We (θ, s)S(θ, s) ur Tzw (θ, s) = ⎣Wu (θ, s)K(θ, s)S(θ, s)⎦. 005 Wy (θ, s)T(θ, s) C The augmented plant G (Figure 1.2) has the two input vectors w and us and the two output vectors z Y and e, where z consists of the three subvectors ze , zu , and zy (Figure 1.3). Its schematic representation is N shown in more detail in Figure 1.4. SU In general, the four subsystems Gs (θ, s), We (θ, s), Wu (θ, s), and Wy (θ, s) are LPV time-invariant systems. By concatenating their individual state vectors into one state vector x, we can describe the dynamics of the augmented plant by the following state–space model: C  w(t) ẋ(t) = A(θ)x(t) + B1 (θ) B2 (θ) us (t) z(t) C (θ) D (θ) D12 (θ) w(t) = 1 x(t) + 11. e(t) C2 (θ) D21 (θ) D22 (θ) us (t) We ze w Wu zu Wy zy us Gs e – FIGURE 1.4 Schematic representation of the augmented plant. i i  i i i i Linear Parameter-Varying Control of Nonlinear Systems 1-5 The following conditions are necessary for the existence of a solution to the H∞ control design problem∗ : 1. The weights We , Wu , and Wy are asymptotically stable. 2. The plant [As , Bs ] is stabilizable. 3. The plant [As , Cs ] is detectable. 4. The maximal singular value of D11 is sufficiently small: σ(D11 ) < γ. 5. Rank(D12 ) = ms , that is, there is a full feedthrough from us to z. 6. Rank(D21 ) = ps = ms , that is, there is a full feedthrough to e from w. 7. The system [A, B1 ] has no uncontrollable poles on the imaginary axis. 8. The system [A, C1 ] has no undetectable poles on the imaginary axis. Remarks Condition 6 is automatically satisfied (see Figure 1.4). Condition 7 demands that the plant Gs has no poles on the imaginary axis. Condition 5 can be most easily satisfied by choosing Wu as a static system with a small, square feedthrough: Wu (s) ≡ εI. In order to present the solution to the H∞ problem in a reasonably esthetic way, it is useful to introduce the following substitutions [3,4]:  um B = B 1 B2  D1 = D11 D12 R̄ = T D − γ2 I D11 11 TD TD D11 TD 12 ul ric D12 11 D12 12 S̄ = BR̄−1 BT ur Ā = A − BR̄−1 D1 T 006 C1 C Q̄ = C1T C1 − C1T D1 R̄−1 D1 T C1 Y G1 G= = R̄−1 (BT K + D1 T C1 ) G2 N 1 T SU R = I − 2 D11 D11 γ R̄¯ = D R(γ)−1 DT 21 21 C 1 C̄¯ = C2 − D21 G1 + 2 D21 R−1 D11 T D12 G2 γ 1 1 S̄¯ = C̄¯ T R̄¯ −1 C̄¯ − 2 G2T D12 T D12 G2 − 4 G2T D12 T D11 R−1 D11 T D12 G2 γ γ 1 Ā¯ = A − B1 G1 + 2 B1 R−1 D11 T T ¯ −1 ¯ D12 G2 − B1 R−1 D21 R̄ C̄ γ ¯ = B R−1 BT − B R−1 DT R̄¯ −1 D R−1 BT. Q̄ 1 1 1 21 21 1 Remarks The matrix K has the same dimension as A. The solution of the first algebraic matrix Riccati equation is given below. The matrix G has the same partitioning as BT. The matrices Q̄ and Q̄¯ will automatically be positive-semidefinite. Remember: all of these matrices are functions of the frozen parameter vector θ. ∗ In order to prevent cumbersome notation, the dependence on the parameter vector θ is dropped in the development of the results. i i  i i i i 1-6 Control System Applications The solution of the H∞ problem is described in the following. Theorem 1.1: The H∞ problem has a solution for a given value γ > 0 if and only if the algebraic matrix Riccati equation 0 = K Ā + ĀT K − K S̄K + Q̄ has a positive-semidefinite stabilizing solution K and if and only if the algebraic matrix Riccati equation ¯ + P Ā¯ T − P S̄P 0 = ĀP ¯ + Q̄ ¯ has a positive-semidefinite stabilizing solution P.  The controller K(s) has the following state–space description: ż(t) = [A − B1 G1 − B2 G2 − H(C2 − D21 G1 − D22 G2 )]z(t) + He(t) us (t) = − G2 z(t) um with the input matrix  H = P C̄¯ T + B1 R−1 D21 T ¯ −1 R̄. ul ric Remark ur For sufficiently large values of γ, both of the algebraic matrix Riccati equations will have unique stabilizing 007 solutions, such that Ā − S̄K and Ā¯ − P S̄¯ are stability matrices. We strive for a solution for γ = 1 for all of C the attainable values of the parameter vector θ. Y N 1.4 Choosing the LPV Weights W (θ, s) SU For a parameter-independent SISO plant Gs (s), a control engineer knows how to choose the bandwidth ωc of the control system (Figure 1.1), that is, the cross-over frequency of the loop gain |Gs ( jω)K( jω)|, C after inspecting the magnitude plot |Gs ( jω)| of the plant. He also knows how to choose quantitative specifications for performance via the sensitivity |S( jω)| in the passband and for robustness via the complementary sensitivity |T( jω)| in the rejection band and the peak values of |S( jω)| (e.g., 3 dB ≈ 1.4) and |T( jω)| (e.g., 1 dB ≈ 1.12) in the crossover region. Since the weights We and Wy shape the sensitivity S and the complementary sensitivity T, respectively, the weights are chosen in the following way: (1) choose the functions S(ω) and T(ω) bounding the quantitative specifications for |S( jω)| and |T( jω)|, respectively, from above, at all frequencies; (2) choose the weights We and Wy as the inverses of S and T, respectively, such that |We ( jω)| ≡ 1/S(ω) and |Wy ( jω)| ≡ 1/T(ω) hold. Example: For the sake of simplicity, let S(ω) correspond to a lead-lag element with S(0) = Smin = 0.01, S(∞) = Smax = 10, and the corner frequencies Smin ωc and Smax ωc ; and let T (ω) correspond to a lag-lead element with T (0) = Tmax = 10, T (∞) = Tmin = 10−3 , and the corner frequencies κωc /Tmax and κωc /Tmin with κ in the range 0.7 < κ ≤ 2... 10. i i  i i i i Linear Parameter-Varying Control of Nonlinear Systems 1-7 For a parameter-dependent SISO plant Gs (θ, s), for each value of the parameter vector θ, we proceed as described above in a coherent way. By coherent, we mean that we should “stress” the plant about equally strongly in each of the attainable operating points. Thus, we should probably have a pretty constant ratio ωc (θ)/ωn (θ) of the chosen bandwidth of the closed-loop control system and the natural bandwidth ωn (θ) of the plant Gs (θ, s). For a SISO plant, we obtain the following weights in the example described above: For We the parameter-varying lag-lead element 1 s + Smax (θ)ωc (θ) We (θ, s) = Smax (θ) s + Smin (θ)ωc (θ) and for Wy (s) the parameter varying lead-lag element κ(θ)ωc (θ) s+ 1 T max (θ) Wy (θ, s) =. T min (θ) κ(θ)ωc (θ) s+ T min (θ) As already mentioned in Section 1.3, the weight Wu is usually chosen as a static element with a very small gain ε (e.g., ε = 10−8 ) in order to automatically satisfy the condition 5: um Wu (θ, s) ≡ ε. In the case of a parameter-varying MIMO plant, the weights We , Wu , and Wy are square matrices ul because e, us , and ys are vectors. In general, we use diagonal matrices with identical diagonal elements ric we (θ, s), wu (θ, s), and wy (θ, s), respectively. If constant parameters Smin , Smax , T max , T min , and κ can be used, this is a good indication that the ur parameter dependence of the cross-over frequency ωc (θ) has been chosen in a coherent way. 008 C 1.5 Handling PV Time Delays Y N In many applications, modeling the linearized plant by a rational transfer matrix Gs (θ, s) only is not SU sufficient because the dynamics of the plant include significant, possibly parameter-varying, time delays Ti (θ). A rational approximation of the transcendent transfer function e−sT of a time delay can be obtained C using the following Padé type approximation [5,7]: N ak (−sT)k (2N − k)! e−sT ≈ k=0 with ak =. N k!(N − k)! ak (sT)k k=0 The coefficients ak can also be calculated recursively with the following scheme: aN = 1 k(2N + 1 − k) ak−1 = ak for k = N,... , 1. N +1−k For choosing the order N of the approximation, we proceed along the following lines of thought : The above Padé approximation is very good at low frequencies. The frequency ω∗ , where the phase error is π/6 (30◦ ), is ω∗ ≈ 2N T. i i  i i i i 1-8 Control System Applications In order to suffer only a small loss of phase margin due to the Padé approximation error, we put ω∗ way out into the rejection band. We propose ω∗ = αωc with α = 30. This leads to the following choice for the approximation order N: α  N(θ) = ωc (θ)T(θ). 2 Obviously, choosing ωc (θ) ∼ 1/T(θ) would yield a parameter-independent approximation order N. 1.6 Applications in Automotive Engine Control In this section, the LPV H∞ methodology for the design of a LPV controller is applied to two problems of fuel control for a 4-stroke, spark-ignited, port-injected gasoline engine: In Section 1.6.1, we discuss the model-based feedback control, and in Section 1.6.2, the model-based feedforward control. As usual, in control, the requirements for the accuracy of the mathematical models upon which the control designs are based differ significantly between feedback control and feedforward control. For the design of a robust feedback control for an asymptotically stable plant, once we have chosen a controller, glibly speaking, it suffices to know the following data with good precision: the crossover um frequency ωc , the (sufficiently large) phase margin ϕ, and the (acceptable) direction of the tangent to the Nyquist curve at ωc. In other words, a rather crude model of the plant just satisfying these requirements will do. ul Conversely, for the design of a feedforward control, a rather precise model of the plant is needed ric because, essentially, the feedforward controller is supposed to invert the dynamics of the plant. The feedback part of the control scheme will then be mainly responsible for stability and robustness, besides ur further improving the command tracking performance. 009 C 1.6.1 Feedback Fuel Control Y N In this example, we want to design a feedback controller. Its task is keeping the air to fuel ratio of the mixture in the cylinders stoichiometric. SU The fuel injection is governed by the control law ti = β(n, m)U. Here, ti is the duration of the injection impulse, n the engine speed, and m the mass of air in a cylinder (calculated by the input manifold observer). C The function β(n, m) is defined in such a way that the dimensionless control variable U is nominally 1 at every static operating point: Unom (n, m) ≡ 1. With a wide-range λ-sensor, the resulting air to fuel ratio is measured in the exhaust manifold of the engine. Its signal Λ is proportional to the air to fuel ratio and scaled such that Λ = 1 corresponds to stoichiometric. Hence, Λnom ≡ 1. We want to find a robust compensator K(θ, s), such that small changes us (t) in the control U(t) = Unom + us (t) will keep the errors λs (t) = Λ(t) − Λnom minimum even in an arbitrary transient operation of the engine. Obviously, the parameter vector describing the operating point is θ(t) = [n(t), m(t)]. For modeling the linearized dynamics of the fuel path of the engine, the following phenomena must be considered: the wall-wetting dynamics in the intake manifold, turbulent mixing of the gas in the exhaust manifold, the dynamics of the λ-sensor, and (last but most important) the time delay between the injection of the fuel and the arrival of the corresponding mixture at the position of the λ-sensor. The simplest model we can get away with successfully is 1 Gs (θ, s) = −e−sT(θ). τ(θ)s + 1 i i  i i i i Linear Parameter-Varying Control of Nonlinear Systems 1-9 The design of the controller proceeds in the following steps: Identify the functions T(θ) for the time delay and τ(θ) for the time constant over the full operating envelope of the engine. This is done by measuring the step response of the engine to a step change in the amount of injected fuel over a sufficiently fine mesh of the parameters θ = [n, m]. Choose a function ωc (T, τ) for the parameter-varying bandwidth of the control system. (Please note the change of variables!) Choose the weighting functions We (T, τ, s), Wu (s), and Wy (T, τ, s). Choose an approximation order N(T) for the Padé approximation of e−sT. This yields a rational approximate transfer function  Gs (T, τ, s) of the plant. Solve the H∞ problem for every pair (T, τ). Reduce the order of the resulting compensator K(T,  τ, s) by one successively, watching the resulting Nyquist curves and stop before the Nyquist curve deforms significantly. For practical purposes, the reduced order of the resulting final compensators should be constant over T and τ. Find a structurally suitable representation for the reduced-order transfer function K(T, τ, s), so that its parameters, say ki (T, τ), can continuously and robustly be mapped over T and τ. The engine control operates as follows in real time: At each control instant, that is, for each upcoming cylinder requesting its injection signal ti , the instantaneous engine speed n and air mass m in the cylinder are available and the following steps are taken: um Calculate T(n, m) and τ(n, m). Calculate the parameters ki (T, τ) of the continuous-time controller with the transfer function K(T, τ, s). Discretize the controller to discrete-time. ul ric Process one time step of the discrete-time controller and output the corresponding signal ti. ur Remark 010 C Note the fine point here: As the engine speed changes, the time increment of the discrete-time controller changes. Y For a BMW 1.8-liter 4-cylinder engine, the time delay T and the time constant τ were found to be in N the ranges T = 0.02... 1.0 s and τ = 0.01... 0.5 s over the full operating envelope of the engine. In , SU the bandwidth was chosen as ωc (T, τ) = π/6T. With α = 30, this resulted in the constant order N = 8 for the Padé approximation of the time delay. For more details about these LPV feedback fuel control schemes, the reader is referred to [4,5,8,11], C and [12, ch. 4.2.2]. 1.6.2 Feedforward Fuel Control In this example, we want to design a feedforward controller. Its task is inverting the wall-wetting dynamics of the intake manifold, such that, theoretically, the air to fuel ratio Λ(t) never deviates from its nominal value Λnom = 1 in dynamic operation of the engine. In 1981, Aquino published an empirical model for the wall-wetting dynamics , that is, for the dynamic mismatch between the mass mFi of fuel injected and the mass mFo of fuel reaching the cylinder:   κ mFo (s) = 1 − κ + mFi (s). sτ + 1 This is a good model in the sense that it captures the balance of the fuel mass flow into the wall-wetting fuel puddle in the intake manifold and the fuel mass flow released by it again. The problem is, that for any given engine, the fraction κ and the time constant τ are strongly dependent on the temperatures of the air, of the fuel, and of the intake manifold wall, and the air mass flow ṁ, the intake manifold pressure, and so on. i i  i i i i 1-10 Control System Applications Therefore, Aquino’s model lends itself to the model-based design of a feedforward fuel controller. But its parameters should be derived using first physical principles. Mathematical models derived by this approach and the corresponding designs of parameter-varying feedforward fuel controllers have been published in [9,10,23–25]. A summary of the model can be found in [12, ch. 2.4.2]. 1.7 Application in Aircraft Flight Control In this section, the LPV H∞ methodology for the design of a LPV controller is applied to the control of the short-period motion of a small unmanned airplane. For controlling an aircraft in a vertical plane, the following two physical control variables are available: The thrust F (for control in the “forward” direction) and the elevator angle δe (for angular control around the pitch axis). In most cases, the airplane’s pitch dynamics can be separated into a fast mode (from the elevator angle δe to the angle of attack α) and a slow mode (from the angle of attack α to the flight path angle γ). Therefore, for flight control, it is more useful to use the angle of attack as a control variable (instead of δe ). In this case, the fast dynamics from δe to α should be considered in the flight control design as actuator dynamics from the commanded angle of attack αcom to the actual angle of attack of the aircraft in a um suitable way. These “actuator dynamics” are usually associated with the notion of “short-period motion” because a step in the elevator angle produces a damped oscillatory response of the angle of attack. Thus, we get the following control problem for the fast inner control loop of the overall control scheme, that is, the problem of controlling the short-period motion: ul For the under-critically damped second-order parameter-varying system with the input δe , the output ric α, and the transfer function Gαδe (θ, s), find a parameter-varying controller with the transfer function ur K(θ, s), such that the command-following system 011 C K(θ, s)Gαδe (θ, s) α(s) = αcom (s) 1 + K(θ, s)Gαδe (θ, s) Y is robust and performs in a satisfactory way over the full operating envelope described by the parameter N vector θ = (v, h) with the velocity v = vmin... vmax and the altitude h = hmin... hmax. SU In [14,15], a small unmanned airplane with a takeoff mass of 28 kg, and a wingspan of 3.1 m, and an operating envelope for the velocity of v = 20... 100 m/s and for the altitude of h = 0... 800 m has been investigated for robust, as well as fail-safe flight control. C The transfer function Gαδe (θ, s) can be written in the form s1 (θ)s2 (θ) Gαδe (θ, s) = Gαδe (θ, 0). (s − s1 (θ))(s − s2 (θ)) Over the full flight envelope, the poles s1 (θ), s2 (θ) (in rad/s) and the steady-state gain Gαδe (θ, 0) can be parametrized with very high precision as follows: s1,2 (v, h) = (c1 + c2 h)v ± j(c3 + c4 h)v, c5 Gαδe (v, h, 0) = 1 + c6 h with c1 = −1.47 × 10−1 rad/m c2 = 1.37 × 10−5 rad/m2 c3 = 7.01 × 10−2 rad/m i i  i i i i Linear Parameter-Varying Control of Nonlinear Systems 1-11 c4 = −3.08 × 10−6 rad/m2 c5 = 1.20 c6 = −7.47 × 10−5 m−1. For manually piloting the unmanned airplane, it is desirable that the dynamic and static charac- teristics of the command following control from αcom to α be parameter-independent over the full flight envelope. This can easily be achieved by using the S/KS/T weighting scheme (Figure 1.3) with parameter-independent weights We , Wu , and Wy. 1 s + Smax (θ)ωc (θ) We (θ, s) = Smax (θ) s + Smin (θ)ωc (θ) κ(θ)ωc (θ) s+ 1 T max (θ) Wy (θ, s) = T min (θ) κ(θ)ωc (θ) s+ T min (θ) Wu (θ, s) ≡ ε. um For the following choice of the constant parameters of the weights: ωc (θ) ≡ 2 rad/s κ(θ) ≡ 1.5 ul ric Smin (θ) ≡ 0.01 Smax (θ) ≡ 10 ur 012 Tmax (θ) ≡ 100 C Tmin (θ) ≡ 0.001 ε(θ) ≡ 10−4 , Y N a suitable unit step response from αcom to α with a rise time of about one second and no overshoot results SU for all v = 20... 100 m/s and h = 0... 800 m. C 1.8 Conclusions In this chapter, some concepts of LPV H∞ control for a LPV plant A(θ), B(θ), C(θ) have been presented in detail. The salient feature is choosing a parameter-varying specification ωc (θ) for the bandwidth of the control system. Furthermore, parameter-varying weighting functions W (θ, s) have been stipulated. Applying these concepts has been discussed briefly for both LPV feedback and LPV feedforward control in the area of fuel control of an automotive engine, as well as for LPV control of the short-period motion in aircraft flight control. In this chapter, the parameter θ has been assumed to be “frozen”, that is, its truly time-varying nature, θ(t), has been neglected. Naturally, the question arises whether such an LVP control system will be asymptotically stable and sufficiently robust even during rapid changes of the parameter vector. Presently, there is a lot of research activity worldwide addressing the question of time-varying parameters, see [16–22] for instance. Suffice it to say here that in our examples about fuel injection of an automotive engine and about the control of the short-period motion of an airplane, the bandwidth of the dynamics of θ(t) is at least an order of magnitude smaller than the bandwidth of our control systems, even in severe transient operation of the engine or the airplane, respectively! i i  i i i i 1-12 Control System Applications References 1. J. C. Doyle, K. Glover, P. P. Khargonekar, and B. A. Francis, State-space solutions to standard H2 and H∞ control problems, IEEE Transactions on Automatic Control, vol. 34, pp. 831–847, 1989. 2. U. Christen, Engineering Aspects of H∞ Control, ETH dissertation no. 11433, Swiss Federal Institute of Technology, Zurich, Switzerland, 1996. 3. H. P. Geering, Robuste Regelung, 3rd ed., IMRT-Press, Institut für Mess- und Regeltechnik, ETH- Zentrum, Zurich, Switzerland, 2004. 4. H. P. Geering and C. A. Roduner, Entwurf robuster Regler mit der H∞ Methode, Bulletin SEV/VSE, no. 3, pp. 55–58, 1999. 5. C. A. Roduner, H∞ -Regelung linearer Systeme mit Totzeiten, ETH dissertation no. 12337, Swiss Federal Institute of Technology, Zurich, Switzerland, 1997. 6. U. Christen, Calibratable model-based controllers, in Proceedings of the IEEE Conference on Control Applications, Glasgow, Scotland, October 2002, pp. 1056–1057. 7. J. Lam, Model reduction of delay systems using Padé approximants, International Journal of Control, vol. 57, no. 2, pp. 377–391, 1993. 8. C. A. Roduner, C. H. Onder, and H. P. Geering, Automated design of an air/fuel controller for an SI engine considering the three-way catalytic converter in the H∞ approach, in Proceedings of the 5th IEEE Mediterranean Conference on Control and Systems, Paphos, Cyprus, July 1997, paper S5-1, pp. 1–7. 9. M. A. Locatelli, Modeling and Compensation of the Fuel Path Dynamics of a Spark Ignited Engine, ETH dissertation no. 15700, Swiss Federal Institute of Technology, Zurich, Switzerland, 2004. 10. M. Locatelli, C. H. Onder, and H. P. Geering, An easily tunable wall-wetting model for port fuel injection engines, in SAE SP-1830: Modeling of Spark Ignition Engines, March 2004, pp. 285–290. um 11. E. Shafai, C. Roduner, and H. P. Geering, Indirect adaptive control of a three-way catalyst, in SAE SP-1149: Electronic Engine Controls, February 1996, pp. 185–193. 12. L. Guzzella and C. H. Onder, Introduction to Modeling and Control of Internal Combustion Engine Systems. London: Springer, 2004. ul ric 13. C. F. Aquino, Transient A/F characteristics of the 5 liter central fuel injection engine, 1981 SAE Inter- national Congress, SAE paper 810494, Detroit, MI, March 1981. 14. M. R. Möckli, Guidance and Control for Aerobatic Maneuvers of an Unmanned Airplane, ETH disserta- ur tion no. 16586, Swiss Federal Institute of Technology, Zurich, Switzerland, 2006. 013 15. G. J. J. Ducard, Fault-Tolerant Flight Control and Guidance Systems for a Small Unmanned Aerial Vehicle, C ETH dissertation no. 17505, Swiss Federal Institute of Technology, Zurich, Switzerland, 2007. 16. J. S. Shamma and M. Athans, Analysis of gain scheduled control for nonlinear plants, IEEE Transactions Y on Automatic Control, vol. 35, pp. 898–907, 1990. 17. R. A. Hyde and K. Glover, The application of scheduled H∞ controllers to a VSTOL aircraft, IEEE N Transactions on Automatic Control, vol. 38, pp. 1021–1039, 1993. SU 18. G. Becker and A. Packard, Robust performance of linear parametrically varying systems using parametrically-dependent linear feedback, Systems & Control Letters, vol. 23, pp. 205–215, 1994. 19. D. A. Lawrence and W. J. Rugh, Gain scheduling dynamic linear controllers for a nonlinear plant, C Automatica, vol. 31, pp. 381–390, 1995. 20. P. Apkarian, P. Gahinet, and G. Becker, Self-scheduled H∞ control of linear parameter-varying systems: A design example, Automatica, vol. 31, pp. 1251–1261, 1995. 21. P. Apkarian and R. J. Adams, Advanced gain-scheduling techniques for uncertain systems, IEEE Trans- actions on Control Systems Technology, vol. 6, pp. 21–32, 1998. 22. F. Bruzelius, Linear Parameter-Varying Systems, Ph.D. dissertation, Chalmers University of Technology, Göteborg, Sweden, 2004. 23. C. H. Onder and H. P. Geering, Measurement of the wall-wetting dynamics of a sequential injection spark ignition engine, in SAE SP-1015: Fuel Systems for Fuel Economy and Emissions, March 1994, pp. 45–51. 24. C. H. Onder, C. A. Roduner, M. R. Simons, and H. P. Geering, Wall-wetting parameters over the operating region of a sequential fuel injected SI engine, in SAE SP-1357: Electronic Engine Controls: Diagnostics and Controls, February 1998, pp. 123–131. 25. M. R. Simons, M. Locatelli, C. H. Onder, and H. P. Geering, A nonlinear wall-wetting model for the complete operating region of a sequential fuel injected SI engine, in SAE SP-1511: Modeling of SI Engines, March 2000, pp. 299–308. i i  i i i i 2 Powertrain Control∗ 2.1 Introduction........................................................ 2-1 Davor Hrovat 2.2 Powertrain Controls and Related Attributes.... 2-3 Ford Motor Company 2.3 Engine Control.................................................... 2-3 Mrdjan Jankovic Fuel-Consumption Optimization Idle Speed Ford Motor Company Control Closed-Loop Air–Fuel Ratio Control 2.4 Transmission Controls..................................... 2-26 Ilya Kolmanovsky 2.5 Drivability.......................................................... 2-30 um Ford Motor Company Tip-In/Back-Out Drivability Cancellation of VCT-Induced Air/Torque Disturbance Stephen Magner 2.6 Diagnostics........................................................ 2-38 Ford Motor Company ul Misfire Detection VCT Monitoring Diana Yanakiev 2.7 Conclusion........................................................ 2-46 ric Ford Motor Company References.................................................................... 2-46 ur 014 2.1 Introduction C Y Automotive controls and associated microcomputers and embedded software represent one of the most active industrial R&D areas. Within this ever expanding area it is no surprise that powertrain controls have N evolved as the most widespread and matured branch since, traditionally, the first microcomputer applica- SU tions started with engine controls. For example, the first four generations of automotive computers found in Ford vehicles during the 1970s and early 1980s were all used for on-board engine controls (Powers, C 1993). The first chassis/suspension and vehicle control applications then followed in the mid-1980s. Early powertrain control applications were driven by U.S. regulatory requirements for improved fuel economy and reduced emissions. Additional benefits included improved functionality, performance, drivability, reliability, and reduced time-to-market as facilitated by the inherent flexibility of comput- ers and associated software. Thus, in the mid-1970s, American automotive manufacturers introduced microprocessor-based engine control systems to meet the sometimes conflicting demands of high fuel economy and low emissions. Present-day engine control systems contain many inputs (e.g., pressures, temperatures, rotational speeds, exhaust gas characteristics) and outputs (e.g., spark timing, exhaust gas recirculation, fuel-injector pulse widths, throttle position, valve/cam timing). The unique aspect of the automotive control is the requirement to develop systems that are relatively low in cost, which will be applied to several hundred thousand units in the field, that must work on automobiles with inherent manufacturing variability, which will be used by a spectrum of human operators, and are subject to irregular maintenance and varying operating conditions. This should be contrasted with the aircraft/spacecraft control problem, for which many of the sophisticated control techniques have been developed. In this case, we can enumerate nearly an opposite set of conditions. ∗ Figures from this chapter are available in color at http://www.crcpress.com/product/isbn/9781420073607 2-1 i i  i i i i 2-2 Control System Applications The software structure of the (embedded) controllers that have been developed to date is much like those in the other areas (i.e., aircraft controllers, process controllers) in that there exists an “outer-loop” operational mode structure, in this case, typically provided by a driver whose commands—for example, gas pedal position—are then interpreted as commands or reference signals for subsequent control agents. The latter typically consist of feedforward–feedback-adaptive (learning) modules. Traditionally, the feed- forward or “open-loop” portion has been by far the most dominant with numerous logical constructs such as “if–then–else” contingency statements and related 2D and 3D tables, myriads of parameters, and online models of underlying physics and devices. More precisely, the feedforward component may include inverse models of relevant components and related physics such as nonlinear static formulas for flow across the throttle actuator, for example. Assuming the existence of the microcomputer module with given chronometric and memory capa- bilities, the structured/disciplined approach to developing a total embedded control system typically involves the following major steps: (1) development of requirements; (2) development of appropriate linear and nonlinear plant models; (3) preliminary design with linear digital control system methods; (4) nonlinear simulation/controller design; and (5) hardware-in-the-loop/real-time simulation capabil- ity for identification, calibration, and verification including confirmation that the above chronometric and memory constraints have not been violated. This includes appropriate dynamometer and vehicle testing with the help of rapid prototyping, autocoding, and data collection and manipulation tools, as needed. Typical powertrain control strategies contain several hundred thousand lines of C code and thou- um sands of associated parameters and calibration variables requiring thousands of man-hours to calibrate, although the ongoing efforts in “self-calibrating” approaches and tools may substantially reduce this ul time-consuming task. In addition, different sensors and actuators—such as Electronic Throttle Con- trol (ETC), Variable Cam Timing (VCT), and Universal Exhaust Gas Oxygen (UEGO) sensors—are ric constantly added and upgraded. So are the new functions and requirements. When computer memory and/or chronometric capabilities have been exhausted and new or improved functionality is needed, this ur 015 can lead to development of new requirements for the next generation of engine computer modules. For C example, in the period 1977–1982, Ford Motor Company introduced four generations of Engine Control Computers (EEC) of ever increasing capabilities. This evolution was needed to address more demand- Y ing fuel economy and pollution requirements, while at the same time for introducing new and/or more N sophisticated functionality and for improved performance (Powers, 1993). The five control strategy development steps mentioned above may not always be sequential and some of SU the steps can be omitted. Moreover, in practice, there are implied iteration loops between different steps. For example, one starts Step 1 with the best available requirements at the time, which can subsequently C be refined as one progresses through Steps 2 through 5. Similarly, the models from Step 2 can be further refined as a result of Step 5 hardware implementation and testing. In some cases, the detailed nonlinear models of a plant or component may already exist and can be used (with possible simplifications) to design a nonlinear plant-based controller directly. Alternatively, the detailed nonlinear model can be used to extract simplified linearized models with details that are relevant for the linear control system design within the bandwidth of interest. The model development, Step 2, may itself include models based on first principles (physics-based models) or semiempirical and identification- based models (“gray” and “black” boxes). Each approach has its advantages and disadvantages. The black and gray box models rely heavily on actual experimental data so that by default they are “validated” and typically require lesser time to develop. On the other hand, physically based models allow for (at least preliminary) controller design even before an actual hardware/plant is built and in the case of open-loop unstable systems they are needed to devise an initial stabilizing controller. Moreover, they can be an invaluable source of insight and critical information about the plant modus operandi, especially when dealing with sometimes elusive dynamic effects. In this context, they can influence the overall system design—both hardware and software—in a true symbiotic way. Before we focus on concrete applications of powertrain controls, it is first important to enumerate what the main drivers and goals of such control systems are. This is briefly summarized in the next section. i i  i i i i Powertrain Control 2-3 2.2 Powertrain Controls and Related Attributes Modern powertrains must satisfy numerous often competing requirements so that the design of a typical powertrain control system involves tradeoffs among a number of attributes (Hrovat and Powers, 1988, 1990). When viewed in a control theory context, the various attributes are categorized quantitatively as follows: Emissions: A set of terminal or final time inequality constraints (e.g., in case of gasoline engine, this would apply to key pollution components: NOx , CO, and HC over certification drive cycles). Fuel consumption: A scalar quantity to be minimized over a drive cycle is usually the objective function to be minimized. Driveability: Expressed as constraints on key characteristic variables such as the damping ratio of dominant vibration modes or as one or more state variable inequality constraints, which must be satisfied at every instant on the time interval (e.g., wheel torque or vehicle acceleration should be within a certain prescribed band). Performance: Either part of the objective function or an intermediate point constraint, for example, achieve a specified 0–60 mph acceleration time. Reliability: As a part of the emission control system, the components in the computer control system (sensors, actuators, and computers) have up to 150,000 mile or 15-year warranty for Partial Zero-Emission Vehicle (PZEV)-certified vehicles. In the design process, reliability can enter as a um sensitivity or robustness condition, for example, location of roots in the complex plane, or more explicitly as uncertainty bounds and weighted sensitivity/complimentary sensitivity bounds in the context of H∞ or μ-synthesis and analysis methodology. ul Cost: The effects of cost are problem dependent. Typical ways that costs enter the problem quanti- ric tatively are increased weights on control variables in quadratic performance indices (which implies relatively lower-cost actuators) and output instead of state feedback (which implies fewer sensors ur but more software). 016 C Packing: Networking of computers and/or smart sensors and actuators requires distributed control theory and tradeoffs among data rates, task partitioning, and redundancy, among others. Y Electromagnetic interference: This is mainly a hardware problem, which is rarely treated explicitly in the analytic control design process. N Tamper-proof: This is one of the reasons for computer control, and leads to adaptive/self-calibrating SU systems so that dealer adjustments are not required as the powertrain ages or changes. To illustrate how control theoretic techniques are employed in the design of powertrain control systems, C the examples of typical engine, transmission, and driveline controls will be reviewed along with some important related considerations such as drivability and diagnostics. This chapter concludes with a discussion of current trends in on-board computer control diagnostics. 2.3 Engine Control Over the last decade, passenger vehicle emission regulations have become stricter by a factor between 5 (based on comparing Tier I and Tier II bin 5 EPA emission standards) and 30 (comparing Tier I and Tier II bin 2, i.e., Super Ultra Low Emission Vehicle, standards). This has forced automakers to increase the exhaust after-treatment catalyst’s size and its precious metal loading, invent new procedures to start a cold engine, introduce a separate cold engine emission-reduction operating mode, and develop a system that accurately controls and coordinates air–fuel ratio, throttle position, spark timing, and VCT. In addition, the system operation has to be monitored by an on-board diagnostic (OBD) system that is designed to respond to any malfunction which can cause an increase in emissions larger than a specified threshold. To improve fuel economy and performance, the automakers have added new engine devices and included new modes of operation. Today’s production engines include ETC and VCT as the standard i i  i i i i 2-4 Control System Applications hardware. Additional devices such as variable valve lift, variable displacement (also called cylinder deacti- vation or displacement-on-demand), charge motion control valves, intake manifold tuning, turbocharg- ers, superchargers, and so on are also used in production applications. Each device operation is computer controlled. Finding the steady-state set-point combinations that achieve the best tradeoff between fuel economy, peak torque/power, and emissions is the subject of the engine mapping and optimization process. Maintaining the device output to the desired set-point typically requires development of a local feedback control system for each one. Engines spend a significant fraction of time in transients. Because the optimization devices may interact in unexpected or undesirable ways, it is important to control and synchronize their transient behavior. Figure 2.1 shows the view of an engine as the system to be controlled. The prominent feature is that the disturbance input, engine speed∗ , and ambient conditions are measured or known, while the performance variables, actual torque, emissions, and fuel efficiency are typically not available (except during laboratory testing). The reference set-point, engine torque demand, is available based on the accelerator pedal position. In the next three sections, we briefly review important paradigms of engine control system design: fuel consumption set-point optimization and feedback regulation. In each case, an advanced optimization or control method has been tried. In each case, experimental tests were run to confirm the achieved benefit. 2.3.1 Fuel-Consumption Optimization um In today’s engines, several optimization devices are added to improve an attribute such as fuel economy. In most cases, the optimal set-point for each device varies with engine operating conditions and is usually ul found experimentally. Combining these devices has made it increasingly difficult and time consuming to map and calibrate such engines. The complexity increases not linearly, but exponentially with the ric number of devices, that is, degrees of freedom. Each additional degree of freedom typically increases the ur complexity in terms of mapping time and size of the calibration tables by a factor between 2 (for two 017 position devices) and 3–10 (for continuously variable devices). For a high-degree-of-freedom (HDOF) C Y Engine speed Torque Disturbance Performance N Ambient condition inputs Emissions outputs SU FE Engine C Throttle Air flow (MAF).. Fuel injectors Measured Engine speed. Control. Spark inputs outputs. A/F. VCT Air temperature........ Engine.. control.. unit Torque demand FIGURE 2.1 The input–output structure of a typical engine control system. ∗ Engine speed can be viewed as a disturbance input in some modes of operation and a system state in others (such as engine idle). i i  i i i i Powertrain Control 2-5 engine, the conventional process that generates the final calibration has become very time consuming. For example, in the dual-independent variable cam timing (diVCT) gasoline engine, which we shall use as the platform for the results in this section, the mapping time could increase by a factor of 30 or more over the conventional (non-VCT) engine or the process could end up sacrificing the potential benefit. From the product development point of view, either outcome is undesirable. The diVCT engine has the intake and exhaust cam actuators that can be varied independently (Jankovic and Magner, 2002; Leone et al., 1996). A typical VCT hardware is shown in Figure 2.2. The intake valve opening (IVO) and exhaust valve closing (EVC) expressed in degrees after top dead center (ATDC) (TDC is at 360 deg crank in Figure 2.2) are considered the two independent degrees of freedom. The experimental 3.0L V6 engine under consideration has the range of −30 to 30 deg ATDC for IVO and 0 to 40 deg ATDC for EVC. 2.3.1.1 Engine Drive Cycle and Pointwise Optimization We consider the problem of optimizing vehicle fuel efficiency while assuring that specific emissions regulations, defined over a drive cycle, are satisfied. A drive cycle, in which the vehicle speed and conditions are specified by regulations, is intended to evaluate vehicle emissions or fuel consumption under a generic um ul ric ur 018 C Y VCT actuator N SU C (Cutaway view) Valve lift Advance Retard Exhaust Intake profile profile 0 EVO 180 IVO 360 EVC 540 IVC 720 Crankshaft TDC BDC TDC BDC TDC angle Power Exhaust Intake Compression FIGURE 2.2 Valve lift profiles versus crank angle in a VCT engine. i i  i i i i 2-6 Control System Applications Vehicle speed (mph) 60 40 20 0 0 50 100 150 200 250 300 350 400 450 500 Engine speed (rpm) 3000 2000 1000 0 50 100 150 200 250 300 350 400 450 500 Engine torque (Nm) 200 100 0 um 0 50 100 150 200 250 300 350 400 450 500 Time (s) ul FIGURE 2.3 Vehicle speed, engine speed, and torque, during the first and last 505 s (bags 1 and 3) of the US75 drive ric cycle. ur 019 or particular drive pattern. The top plot in Figure 2.3 shows the vehicle speed profile during a part of the C US75 drive cycle. Given the transmission shift schedule, the vehicle speed uniquely determines the engine speed and torque needed to follow the drive trace. Hence, from the point of view of engine optimization, Y engine speed (middle plot) and engine torque (bottom plot) are constrained variables. N Due to the presence of three-way catalysts, which become very efficient in removing regulated exhaust SU gases after light-off (50–100 s from a cold start), this optimization problem basically splits into two disjoint problems: C Achieve a fast catalyst light-off while managing feedgas (engine-out) emissions. Optimize fuel economy after catalyst light-off. In this section, we shall only consider the latter problem. Given that the engine speed and torque are constrained by the drive cycle, and the air–fuel ratio is kept close to stoichiometry to assure high efficiency of the catalyst system, the variables one can use to optimize fuel consumption are IVO, EVC, and the spark timing (spk). Basically, we are looking for the best combination of these three variables at various speed and torque points of engine operation (see Figure 2.3). As the fuel consumption is evaluated at a fixed torque, it is often referred to as the brake- specific fuel consumption (BSFC). Figure 2.4 shows BSFC versus two optimization variables at a typical speed/torque operating point (1500 rpm engine speed, 62 Nm torque). The top plot shows BSFC versus IVO and EVC at the spark timing for best fuel economy (called maximum brake torque (MBT) spark). The bottom plot shows BSFC versus IVO and spark, at EVC = 30◦. Hence, the dash curves in Figure 2.4a and b show the same set of mapped points. The plots have been obtained by the full-factorial mapping, which is prohibitively time consuming to generate. On the other hand, achieving the fuel economy potential of the diVCT technology requires accurate knowledge of these (hyper)surfaces under all operating conditions. i i  i i i i Powertrain Control 2-7 (a) 0.345 0.34 BSFC (kg/kWh) 0.335 0.33 0.325 0.32 0.315 0 –30 –20 10 –10 20 0 10 IVO EVC 30 20 40 30 um (b)